How do you clear a player's backpack/inventory completely in a local script?

you can just use the ClearAllChildren function to get rid of every tool in the backpack, like so

game.Players.LocalPlayer.Backpack:ClearAllChildren()

however, the player might also have a tool equipped, so if a tool is in their character, you'll need to destroy that too

local toolInCharacter = game.Players.LocalPlayer.Character:FindFirstChildWhichIsA("Tool")
if toolInCharacter then -- make sure it exists before destroying it
    toolInCharacter:Destroy()
end

in your script that would look like this

local Players = game:GetService("Players") -- it's good practice to use GetService instead of directly indexing the game
local ReplicatedStorage = game:GetService("ReplicatedStorage")
local SoundService = game:GetService("SoundService")
local LocalPlayer = Players.LocalPlayer -- you reference the localplayer a lot, so it's good to use a variable instead

script.Parent.MouseButton1Click:Connect(function() --Click detector
    -- clear tools
    LocalPlayer.Backpack:ClearAllChildren()
    local toolInCharacter = LocalPlayer.Character:FindFirstChildWhichIsA("Tool")
    if toolInCharacter then -- make sure it exists before destroying it
        toolInCharacter:Destroy()
    end
    -- clone the new guns
    ReplicatedStorage.Shotguns.Shotgun:Clone().Parent = LocalPlayer.Backpack
    ReplicatedStorage.Pistols.Pistol3:Clone().Parent = LocalPlayer.Backpack
    script.Parent.Parent.Visible = false
    SoundService.Barret_Equip:Play() -- in most cases you should use the :Play function instead of the .Playing property
end)

I'll try and answer this. Basically you will need to get the children of your backpack, like so:

ItemsInBackPack = game.Players.LocalPlayer.Backpack:GetChildren()

Then you will need to destroy the children:

ItemsInBackPack = game.Players.LocalPlayer.Backpack:GetChildren()
ItemsInBackPack:Destroy()
-- You don't need to use variables for the backpack's children, just making it more clear.

It's quite simple, here is your full code:

script.Parent.MouseButton1Click:Connect(function() --Click detector
    ItemsInBackPack = game.Players.LocalPlayer.Backpack:GetChildren()
    ItemsInBackPack:Destroy()

        game.ReplicatedStorage.Shotguns.Shotgun:Clone().Parent =
         game.Players.LocalPlayer.Backpack --Shotgun Giver
        game.ReplicatedStorage.Pistols.Pistol3:Clone().Parent =
         game.Players.LocalPlayer.Backpack --Pistol Giver
        script.Parent.Parent.Visible = false --Sets the frame status to invisble
        game.SoundService.Barret_Equip.Playing = true --Plays reload sound effect
    end) --Note this is a *local script*

I am not sure if this will work on a localscript, but you will have to try for yourself, happy to help!

If you have any other things you need add me on Discord: Haunted#5196

Use the :ClearAllChildren function.

script.Parent.MouseButton1Click:Connect(function() --Click detector
    game.Players.LocalPlayer.Backpack:ClearAllChildren()

        game.ReplicatedStorage.Shotguns.Shotgun:Clone().Parent =
         game.Players.LocalPlayer.Backpack --Shotgun Giver
        game.ReplicatedStorage.Pistols.Pistol3:Clone().Parent =
         game.Players.LocalPlayer.Backpack --Pistol Giver
        script.Parent.Parent.Visible = false --Sets the frame status to invisble
        game.SoundService.Barret_Equip.Playing = true --Plays reload sound effect
    end) --Note this is a *local script*